{"status": "success", "data": {"description_md": "A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$  What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?<br>\n\n$\\textbf{(A) } \\frac{1}{4} \\qquad\\textbf{(B) } \\frac{2}{7} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{3}{8} \\qquad\\textbf{(E) } \\frac{3}{7}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin  <span class=\"katex--inline\">k</span>  is  <span class=\"katex--inline\">2^{-k}</span>  for  <span class=\"katex--inline\">k = 1,2,3....</span>   What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?<br/></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } \\frac{1}{4} \\qquad\\textbf{(B) } \\frac{2}{7} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{3}{8} \\qquad\\textbf{(E) } \\frac{3}{7}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2019 AMC 12B Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12B_p14", "prev": "/problem/19_amc12B_p12"}}